Why is abiding important in the daily life of a person?

Have to be careful to describe abiding in a terminology people can understand, and not into phrases that many may not fully understand. 

How can I have a relationship that is measurable as your (Peter’s) relationship was with Christ? 1 John 3

If what John said was true, it is everything for us. 

How does this change my thought process, my emotions, my perceptions?

John 14, shows us this relationship with a comparison between him and the Father, and Him and us. 

Through our relationship with the Lord, there is a hope for deliverance.

Psalm 147

Companionship with God, is what Abiding means during the hard times. He can be the hope we fall back on during times of trouble.